In the present work, we study the effects of the long range Coulomb interaction on a second-order DSM.
The Coulomb interaction is ubiquitous in condensed matter systems. Especially, its long range nature
remains intact in semimetals, in contrast with the Fermi liquid. For the 3D first-order DSM at weak
coupling, one-loop renormalization group (RG) analysis indicates that the Fermi velocity grows to infinity
logarithmically at low energies, leading to the marginally irrelevant dimensionless coupling constant[1]}, [2]}, [3]}.
For the second-order DSM, the full rotation symmetry is lost even in the continuum limit, and thus the
low-energy physics is described by two parameters: the dimensionless coupling constant and the ratio of
the anisotropy parameters. We employ the method of RG and show that the coupling constant flows to zero
at low energies, similar to the first-order DSM, where as the ratio of the ansiotropy parameters flows to
a fixed value at low energies. To illustrate the effects of the Coulomb interaction, we calculate the
temperature dependence of the specific heat in terms of the RG equations. We find that in comparison
with the specific heat of a non-interacting second-order DSM, the Coulomb interaction suppresses its
value at all temperatures, except at zero temperature and the temperature close to the band width.
